Question

A gun emplacement is on the edge of a cliff of height h. Prove that the greatest horizontal distance at which a shell from the emplacement can hit the ship is 2(k(k+h))^1/2. And the horizontal distance at which a gun in the ship can hit the placement is 2(k(k-h))^1/2. Where k>h. The initial firing speed of a shell in each case is (2gk)^1/2

Answer 1

a) y equation
0 = h + (2gk)^(1/2) sin theta t - 1/2 g t^2
x equation
R = ( 2 g k)^(1/2) cos theta t
t = R/( ( 2 g k)^(1/2) cos theta)

plug into y equations
0 = h + R tan(theta) - 1/2 g R^2/( 2 g k cos^2 theta)
0 = h + R tan(theta) - R^2/( 4 k cos theta^2)

max range at theta = 45

0 = h + R - R^2/2 k
0 = 2 k h + 2k R - R^2
R^2 - 2kR - 2khh = 0

quadratic equations
R = 2 (k(k+h))^1/2